In several important and active fields of modern applied
mathematics, such as the numerical solution of PDE-constrained
control problems or various applications in image processing and
data fitting, the evaluation of (integer and real) Sobolev norms
constitutes a crucial ingredient. Different approaches exist for
varying ranges of smoothness indices and with varying properties
concerning exactness, equivalence and the computing time for the
numerical evaluation. These can usually be expressed in terms of
discrete Riesz operators.
We propose a collection of criteria which allow to compare different
constructions. Then we develop a unified approach which is valid
for non-negative real smoothness indices for standard finite
elements, and for positive and negative real smoothness for
biorthogonal wavelet bases. This construction delivers a wider
range of exactness than the currently known constructions and is
computable in linear time.
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